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    "# 概率\n",
    ":label:`sec_prob`\n",
    "\n",
    "在某种形式上，机器学习就是做出预测。\n",
    "\n",
    "根据病人的临床病史，我们可能想预测他们在下一年心脏病发作的*概率*。在异常检测中，我们可能想要评估飞机喷气发动机的一组读数是正常运行情况的*可能性*有多大。在强化学习中，我们希望智能体（agent）能在一个环境中智能地行动。这意味着我们需要考虑在每种可行的行为下获得高奖励的概率。当我们建立推荐系统时，我们也需要考虑概率。例如，假设我们为一家大型在线书店工作。我们可能希望估计特定用户购买特定图书的概率。为此，我们需要使用概率学。\n",
    "有完整的课程、专业、论文、职业、甚至院系，都致力于概率学的工作。所以很自然地，我们在这部分的目标不是教授整个科目。相反，我们希望让你起步，教给你足够的知识，使你能够开始构建你的第一个深度学习模型，并让你对该主题有足够的了解，以便你可以开始自己探索它。\n",
    "\n",
    "在前面的章节中，我们已经提到了概率，但没有明确说明它们是什么，也没有给出具体的例子。现在让我们更认真地考虑第一个例子：根据照片区分猫和狗。这听起来可能很简单，但实际上是一个艰巨的挑战。首先，问题的难度可能取决于图像的分辨率。\n",
    "\n",
    "![不同分辨率的图像 ($10 \\times 10$, $20 \\times 20$, $40 \\times 40$, $80 \\times 80$, 和 $160 \\times 160$ pixels).](../img/cat-dog-pixels.png)\n",
    ":width:`300px`\n",
    ":label:`fig_cat_dog`\n",
    "\n",
    "如 :numref:`fig_cat_dog` 所示，虽然人类很容易以 $160 \\times 160$ 像素的分辨率识别猫和狗，但它在 $40 \\times 40$ 像素上变得具有挑战性，而且在 $10 \\times 10$ 像素下几乎是不可能的。换句话说，我们在很远的距离（从而降低分辨率）区分猫和狗的能力可能会接近不知情的猜测。\n",
    "概率给了我们一种正式的途径来说明我们的确定性水平。\n",
    "如果我们完全肯定图像是一只猫，我们说标签$y$是\"猫\"的*概率*，表示为$P(y=$ \"猫\"$)$ 等于 $1$。如果我们没有证据表明 $y =$ “猫” 或 $y =$ “狗”，那么我们可以说这两种可能性是等可能的，把它表示为$P(y=$ \"猫\"$) = P(y=$ \"狗\"$) = 0.5$。\n",
    "如果我们有足够的信心，但不确定图像描绘的是一只猫，我们可以将概率赋值为$0.5 < P(y=$ \"猫\"$) < 1$。\n",
    "\n",
    "现在考虑第二个例子：给出一些天气监测数据，我们想预测明天北京下雨的概率。如果是夏天，下雨的概率是0.5。\n",
    "\n",
    "在这两种情况下，我们都不确定结果。但这两种情况之间有一个关键区别。在第一种情况中，图像实际上是狗或猫，我们只是不知道哪个。在第二种情况下，结果实际上可能是一个随机的事件（如果你相信这些东西。大多数物理学家都相信）。因此，概率是一种灵活的语言，用于说明我们的确定程度，并且它可以有效地应用于广泛的上下文中。\n",
    "\n",
    "## 基本概率论\n",
    "\n",
    "假设我们掷骰子，想知道看到1的几率有多大，而不是看到另一个数字。如果骰子是公平的，那么所有六个结果$\\{1, \\ldots, 6\\}$都有相同的可能发生，因此我们将在每六次中看到一个$1$。我们可以说$1$发生的概率为$\\frac{1}{6}$。\n",
    "\n",
    "对于我们从工厂收到的真实骰子，我们可能不知道那些比例，我们需要检查它是否有污染。调查骰子的唯一方法是多次投掷并记录结果。对于每个骰子，我们将观察到 $\\{1, \\ldots, 6\\}$ 中的一个值。给定这些结果，我们想调查每个结果的概率。\n",
    "\n",
    "对于每个值，一种自然的方法是将单个计数的值除以投掷的总次数。\n",
    "这给了我们一个给定*事件*的概率的*估计值*。*大数定律*（law of large numbers）告诉我们，随着投掷次数的增加，这个估计值会越来越接近真实的潜在概率。在深入了解这里的细节之前，让我们先试一试。\n",
    "\n",
    "首先，让我们导入必要的软件包。\n"
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   "source": [
    "%matplotlib inline\n",
    "import torch\n",
    "from torch.distributions import multinomial\n",
    "from d2l import torch as d2l\n"
   ]
  },
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    "接下来，我们将希望能够投掷骰子。在统计学中，我们把从概率分布中抽取样本的过程称为*抽样*（sampling）。\n",
    "将概率分配给一些离散选择的分布称为*多项分布*（multinomial distribution）。稍后我们将给出*分布*（distribution）的更正式定义。但笼统来说，可以把它看作是对事件的概率分配。\n",
    "\n",
    "为了抽取一个样本，我们只需传入一个概率向量。\n",
    "输出是另一个相同长度的向量：它在索引$i$处的值是采样结果对应于$i$的次数。\n"
   ]
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   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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    "tab": [
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    {
     "data": {
      "text/plain": [
       "tensor([0., 0., 1., 0., 0., 0.])"
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   "source": [
    "fair_probs = torch.ones([6]) / 6\n",
    "multinomial.Multinomial(1, fair_probs).sample()"
   ]
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   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 8
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   "source": [
    "如果你运行采样器很多次，你会发现每次你都得到随机的值。在估计一个骰子的公平性时，我们经常希望从同一分布中生成多个样本。如果用Python的for循环来完成这个任务，速度会慢得令人难以忍受，因此我们使用的函数支持同时抽取多个样本，返回我们想要的任意形状的独立样本数组。\n"
   ]
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   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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    "tab": [
     "pytorch"
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    {
     "data": {
      "text/plain": [
       "tensor([1., 1., 2., 0., 4., 2.])"
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     "execution_count": 3,
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   ],
   "source": [
    "multinomial.Multinomial(10, fair_probs).sample()"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 12
   },
   "source": [
    "现在我们知道如何对骰子进行采样，我们可以模拟1000次投掷。然后，我们可以统计1000次投掷后, 每个数字被投中了多少次。具体来说，我们计算相对频率作为真实概率的估计。\n"
   ]
  },
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   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "origin_pos": 14,
    "tab": [
     "pytorch"
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    {
     "data": {
      "text/plain": [
       "tensor([0.1680, 0.1670, 0.1660, 0.1830, 0.1520, 0.1640])"
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   "source": [
    "# 将结果存储为32位浮点数以进行除法\n",
    "counts = multinomial.Multinomial(1000, fair_probs).sample()\n",
    "counts / 1000  # 相对频率作为估计值"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "因为我们是从一个公平的骰子中生成的数据，我们知道每个结果都有真实的概率$\\frac{1}{6}$，大约是$0.167$，所以上面输出的估计值看起来不错。\n",
    "\n",
    "我们也可以看到这些概率如何随着时间的推移收敛到真实概率。让我们进行500组实验，每组抽取10个样本。\n"
   ]
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   "metadata": {
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   "source": [
    "counts = multinomial.Multinomial(10, fair_probs).sample((500,))\n",
    "cum_counts = counts.cumsum(dim=0)\n",
    "estimates = cum_counts / cum_counts.sum(dim=1, keepdims=True)\n",
    "\n",
    "d2l.set_figsize((6, 4.5))\n",
    "for i in range(6):\n",
    "    d2l.plt.plot(estimates[:, i].numpy(), label=(\"P(die=\" + str(i + 1) + \")\"))\n",
    "d2l.plt.axhline(y=0.167, color='black', linestyle='dashed')\n",
    "d2l.plt.gca().set_xlabel('Groups of experiments')\n",
    "d2l.plt.gca().set_ylabel('Estimated probability')\n",
    "d2l.plt.legend();"
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    "每条实线对应于骰子的6个值中的一个，并给出骰子在每组实验后出现值的估计概率。\n",
    "当我们通过更多的实验获得更多的数据时，这$6$条实体曲线向真实概率收敛。\n",
    "\n",
    "### 概率论公理\n",
    "\n",
    "在处理骰子掷出时，我们将集合 $\\mathcal{S} = \\{1, 2, 3, 4, 5, 6\\}$ 称为 *样本空间*（sample space） 或 *结果空间*（outcome space），其中每个元素都是 *结果*（outcome）。*事件*（event） 是来自给定样本空间的一组结果。例如，“看到 $5$”（$\\{5\\}$）和 “看到奇数”（$\\{1, 3, 5\\}$）都是掷出骰子的有效事件。注意，如果随机实验的结果在事件 $\\mathcal{A}$ 中，则事件 $\\mathcal{A}$ 已经发生。也就是说，如果投掷出$3$点，因为 $3 \\in \\{1, 3, 5\\}$ ，我们可以说，“看到奇数” 的事件发生了。\n",
    "\n",
    "形式上，*概率*（probability） 可以被认为是将集合映射到真实值的函数。在给定的样本空间 $\\mathcal{S}$ 中，事件$\\mathcal{A}$的概率，表示为 $P(\\mathcal{A})$，满足以下属性：\n",
    "\n",
    "* 对于任意事件 $\\mathcal{A}$，其概率从不会是负数，即 $P(\\mathcal{A}) \\geq 0$；\n",
    "* 整个样本空间的概率为 $1$，即 $P(\\mathcal{S}) = 1$；\n",
    "* 对于任意事件 $\\mathcal{A}_1, \\mathcal{A}_2, \\ldots$ 的可数序列，这些事件*互斥*（mutually exclusive）（对于所有 $i \\neq j$ 都有 $\\mathcal{A}_i \\cap \\mathcal{A}_j = \\emptyset$），任何事件发生的概率等于它们各自发生的概率之和，即 $P(\\bigcup_{i=1}^{\\infty} \\mathcal{A}_i) = \\sum_{i=1}^{\\infty} P(\\mathcal{A}_i)$。\n",
    "\n",
    "这些也是概率论的公理，由科尔莫戈罗夫于 1933 年提出。有了这个公理系统，我们可以避免任何关于随机性的哲学争论；相反，我们可以用数学语言严格地推理。例如，让事件 $\\mathcal{A}_1$ 为整个样本空间，且当所有$i > 1$时的$\\mathcal{A}_i = \\emptyset$，我们可以证明 $P(\\emptyset) = 0$，即不可能发生事件的概率是 $0$。\n",
    "\n",
    "### 随机变量\n",
    "\n",
    "在我们掷骰子的随机实验中，我们引入了 *随机变量*（random variable） 的概念。随机变量几乎可以是任何数量，并且不是确定性的。它可以在随机实验的一组可能性中取一个值。考虑一个随机变量 $X$，其值在掷骰子的样本空间 $\\mathcal{S} = \\{1, 2, 3, 4, 5, 6\\}$ 中。我们可以将事件 “看到一个 $5$” 表示为 $\\{X = 5\\}$ 或 $X = 5$，其概率表示为 $P(\\{X = 5\\})$ 或 $P(X = 5)$。通过 $P(X = a)$，我们区分了随机变量 $X$ 和 $X$ 可以采取的值（例如 $a$）。然而，这可能会导致繁琐的表示。\n",
    "为了简化符号，一方面，我们可以将 $P(X)$ 表示为随机变量 $X$ 上的 *分布*（distribution）：分布告诉我们 $X$ 获得任意值的概率。另一方面，我们可以简单用 $P(a)$ 表示随机变量取值 $a$ 的概率。由于概率论中的事件是来自样本空间的一组结果，因此我们可以为随机变量指定值的可取范围。例如，$P(1 \\leq X \\leq 3)$ 表示事件的概率 $\\{1 \\leq X \\leq 3\\}$，这意味着 $\\{X = 1, 2, \\text{or}, 3\\}$。等价地，$P(1 \\leq X \\leq 3)$ 表示随机变量 $X$ 从 $\\{1, 2, 3\\}$ 中取值的概率。\n",
    "\n",
    "请注意，*离散* (discrete) 随机变量（如骰子的侧面）和 *连续* (continuous) 变量（如人的体重和身高）之间存在微妙的区别。问两个人是否具有完全相同的身高没有什么意义。如果我们进行足够精确的测量，你会发现这个星球上没有两个人具有完全相同的身高。事实上，如果我们采取足够精细的测量，在你起床和去睡觉时都不会得到相同的身高。因此，问一个人身高为 1.80139278297192196202 米高的概率是没有任何意义的。考虑到世界上的人口数量，这个概率几乎是 0。在这种情况下，询问某人的身高是否落入给定的区间，比如是否在 1.79 米和 1.81 米之间更有意义。在这些情况下，我们将这个看到某个数值的可能性量化为 *密度* (density)。恰好 1.80 米的高度上没有概率，但密度不是 0。在任何两个不同高度之间的区间，我们都有非零的概率。在本节的其余部分中，我们将考虑离散空间中的概率。对于连续随机变量的概率，您可以参考 :numref:`sec_random_variables`。\n",
    "\n",
    "## 处理多个随机变量\n",
    "\n",
    "很多时候，我们会希望一次考虑多个随机变量。比如，我们可能需要对疾病和症状之间的关系进行建模。给定一个疾病和一个症状，比如 “流感” 和 “咳嗽”，会在某个患者身上，以某个概率存在或不存在关系。虽然我们可能希望这两者发生的概率都接近于零，但我们可能需要估计这些概率和它们之间的关系，以便我们可以运用我们的推断来实现更好的医疗服务。\n",
    "\n",
    "再举一个更复杂的例子：图像包含数百万像素，因此有数百万个随机变量。在许多情况下，图像会附带一个标签，标识图像中的对象。我们也可以将标签视为一个随机变量。我们甚至可以将所有元数据视为随机变量，例如位置、时间、光圈、焦距、ISO、对焦距离和相机类型。所有这些都是联合发生的随机变量。当我们处理多个随机变量时，会有若干个变量是我们感兴趣的。\n",
    "\n",
    "### 联合概率\n",
    "\n",
    "第一个被称为 *联合概率* (joint probability) $P(A = a, B=b)$。给定任何值 $a$ 和 $b$, 联合概率可以回答,  $A=a$ 和 $B=b$ 同时满足的概率是多少? 请注意，对于任何值，对于任何 $a$ 和 $b$ 的取值，$P(A = a, B=b) \\leq P(A=a)$。这点是确定的，因为要同时发生 $A=a$ 和 $B=b$，$A=a$就必须发生，$B=b$也必须发生（反之亦然）。因此，$A=a$ 和 $B=b$ 同时发生的可能性不大于 $A=a$ 或是 $B=b$ 的可能性。\n",
    "\n",
    "### 条件概率\n",
    "\n",
    "这给我们带来了一个有趣的比率：$0 \\leq \\frac{P(A=a, B=b)}{P(A=a)} \\leq 1$。我们称这个比率为 *条件概率* (conditional probability)，并用 $P(B=b \\mid A=a)$ 表示它：它是 $B=b$ 的概率，前提是发生了 $A=a$。\n",
    "\n",
    "### 贝叶斯定理\n",
    "\n",
    "使用条件概率的定义，我们可以得出统计数据中最有用和最著名的方程之一：*Bayes 定理* (Bayes' theorem)。它如下所示。通过构造，我们有 *乘法规则*， $P(A, B) = P(B \\mid A) P(A)$。根据对称性，这也适用于 $P(A, B) = P(A \\mid B) P(B)$。假设 $P(B)>0$， 求解其中一个条件变量，我们得到\n",
    "\n",
    "$$P(A \\mid B) = \\frac{P(B \\mid A) P(A)}{P(B)}.$$\n",
    "\n",
    "请注意，在这里我们使用更紧凑的表示法，其中 $P(A, B)$ 是一个 *联合分布*，$P(A \\mid B)$ 是一个 *条件分布*。这种分布可以在给定值 $A = a, B=b$ 上进行求值。\n",
    "\n",
    "###  边际化\n",
    "\n",
    "如果我们想从另一件事中推断一件事，但我们只知道相反方向的属性，比如因和果的时候，Bayes 定理是非常有用的，正如我们将在本节后面看到的那样。为了能进行这项工作，我们需要的一个重要操作是 *边际化*。这项操作是从 $P(A, B)$ 中确定 $P(B)$ 的操作。我们可以看到，$B$ 的概率相当于计算 $A$ 的所有可能选择，并将所有选择的联合概率聚合在一起：\n",
    "\n",
    "$$P(B) = \\sum_{A} P(A, B),$$\n",
    "\n",
    "这也称为 *求和规则*。边际化结果的概率或分布称为 *边际概率* 或 *边际分布*。\n",
    "\n",
    "### 独立性\n",
    "\n",
    "另一个要检查的有用属性是 *依赖* 与 *独立*。两个随机变量 $A$ 和 $B$ 是独立的，意味着事件 $A$ 的发生不会透露有关 $B$ 事件的发生情况的任何信息。在这种情况下，统计学家通常将这一点表述为 $A \\perp  B$。根据贝叶斯定理，马上就能同样得到 $P(A \\mid B) = P(A)$。在所有其他情况下，我们称 $A$ 和 $B$ 依赖。比如，一个骰子的两次连续抛出是独立的。相比之下，灯开关的位置和房间的亮度并不是（尽管它们不是具有确定性的，因为总是可能存在灯泡坏掉，电源故障，或者开关故障）。\n",
    "\n",
    "由于 $P(A \\mid B) = \\frac{P(A, B)}{P(B)} = P(A)$ 等价于 $P(A, B) = P(A)P(B)$，因此两个随机变量是独立的当且仅当两个随机变量的联合分布是其各自分布的乘积。同样地，给定另一个随机变量 $C$时，两个随机变量 $A$ 和 $B$ 是 *条件独立的* ，当且仅当 $P(A, B \\mid C) = P(A \\mid C)P(B \\mid C)$ 。这个情况表示为 $A \\perp B \\mid C$。\n",
    "\n",
    "### 应用\n",
    ":label:`subsec_probability_hiv_app`\n",
    "\n",
    "让我们用实战考验一下我们的技能。假设一个医生对患者进行艾滋病病毒（HIV）测试。这个测试是相当准确的，如果患者健康但测试显示他患病，这样的失败概率只有 1% 。此外，如果患者真正感染HIV，它永远不会检测不出。我们使用 $D_1$ 来表示诊断结果（如果阳性，则为 $1$，如果阴性，则为 $0$），$H$ 来表示感染艾滋病病毒的状态（如果阳性，则为 $1$，如果阴性，则为0）。\n",
    "在 :numref:`conditional_prob_D1` 中列出了这样的条件概率。\n",
    "\n",
    ":条件概率为 $P(D_1 \\mid H)$。\n",
    "\n",
    "| 条件概率 | $H=1$ | $H=0$ |\n",
    "|---|---|---|\n",
    "|$P(D_1 = 1 \\mid H)$|            1 |         0.01 |\n",
    "|$P(D_1 = 0 \\mid H)$|            0 |         0.99 |\n",
    ":label:`conditional_prob_D1`\n",
    "\n",
    "请注意，每列的加和都是 1（但每行的加和不是），因为条件概率需要总和为1，就像概率一样。让我们计算如果测试出来呈阳性，患者感染HIV的概率，即 $P(H = 1 \\mid D_1 = 1)$。显然，这将取决于疾病有多常见，因为它会影响错误警报的数量。假设人口总体是相当健康的，例如，$P(H=1) = 0.0015$。为了应用贝叶斯定理，我们需要运用边际化和乘法规则来确定\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(D_1 = 1) \\\\\n",
    "=& P(D_1=1, H=0) + P(D_1=1, H=1)  \\\\\n",
    "=& P(D_1=1 \\mid H=0) P(H=0) + P(D_1=1 \\mid H=1) P(H=1) \\\\\n",
    "=& 0.011485.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "因此，我们得到\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(H = 1 \\mid D_1 = 1)\\\\ =& \\frac{P(D_1=1 \\mid H=1) P(H=1)}{P(D_1=1)} \\\\ =& 0.1306 \\end{aligned}.$$\n",
    "\n",
    "换句话说，尽管使用了非常准确的测试，患者实际上患有艾滋病的几率只有 13.06%。正如我们所看到的，概率可能是违反直觉的。\n",
    "\n",
    "患者在收到这样可怕的消息后应该怎么办？很可能，患者会要求医生进行另一次测试来了解清楚。第二个测试具有不同的特性，它不如第一个测试那么好，如 :numref:`conditional_prob_D2` 所示。\n",
    "\n",
    ":条件概率为 $P(D_2 \\mid H)$。\n",
    "\n",
    "| 条件概率 | $H=1$ | $H=0$ |\n",
    "|---|---|---|\n",
    "|$P(D_2 = 1 \\mid H)$|            0.98 |         0.03 |\n",
    "|$P(D_2 = 0 \\mid H)$|            0.02 |         0.97 |\n",
    ":label:`conditional_prob_D2`\n",
    "\n",
    "不幸的是，第二次测试也显示阳性。让我们通过假设条件独立性来计算出应用 Bayes 定理的必要概率：\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(D_1 = 1, D_2 = 1 \\mid H = 0) \\\\\n",
    "=& P(D_1 = 1 \\mid H = 0) P(D_2 = 1 \\mid H = 0)  \\\\\n",
    "=& 0.0003,\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(D_1 = 1, D_2 = 1 \\mid H = 1) \\\\\n",
    "=& P(D_1 = 1 \\mid H = 1) P(D_2 = 1 \\mid H = 1)  \\\\\n",
    "=& 0.98.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "现在我们可以应用边际化和乘法规则：\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(D_1 = 1, D_2 = 1) \\\\\n",
    "=& P(D_1 = 1, D_2 = 1, H = 0) + P(D_1 = 1, D_2 = 1, H = 1)  \\\\\n",
    "=& P(D_1 = 1, D_2 = 1 \\mid H = 0)P(H=0) + P(D_1 = 1, D_2 = 1 \\mid H = 1)P(H=1)\\\\\n",
    "=& 0.00176955.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "最后，鉴于存在两次阳性检测，患者患有艾滋病的概率为\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(H = 1 \\mid D_1 = 1, D_2 = 1)\\\\\n",
    "=& \\frac{P(D_1 = 1, D_2 = 1 \\mid H=1) P(H=1)}{P(D_1 = 1, D_2 = 1)} \\\\\n",
    "=& 0.8307.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "也就是说，第二次测试使我们能够对患病的情况获得更高的信心。尽管第二次检验比第一次检验的准确性要低得多，但它仍然显著改善了我们的估计。\n",
    "\n",
    "## 期望和差异\n",
    "\n",
    "为了概括概率分布的关键特征，我们需要一些测量方法。随机变量 $X$ 的 *期望*（或平均值）表示为\n",
    "\n",
    "$$E[X] = \\sum_{x} x P(X = x).$$\n",
    "\n",
    "当函数 $f(x)$ 的输入是从分布 $P$ 中抽取的随机变量时，$f(x)$ 的期望值为\n",
    "\n",
    "$$E_{x \\sim P}[f(x)] = \\sum_x f(x) P(x).$$\n",
    "\n",
    "在许多情况下，我们希望衡量随机变量 $X$ 与其期望值的偏置。这可以通过方差来量化\n",
    "\n",
    "$$\\mathrm{Var}[X] = E\\left[(X - E[X])^2\\right] =\n",
    "E[X^2] - E[X]^2.$$\n",
    "\n",
    "它的平方根被称为 *标准差* (standared deviation)。随机变量函数的方差衡量的是，当从该随机变量分布中采样不同值 $x$ 时，函数值偏离该函数的期望的程度：\n",
    "\n",
    "$$\\mathrm{Var}[f(x)] = E\\left[\\left(f(x) - E[f(x)]\\right)^2\\right].$$\n",
    "\n",
    "## 小结\n",
    "\n",
    "* 我们可以从概率分布中采样。\n",
    "* 我们可以使用联合分布、条件分布、Bayes 定理、边缘化和独立性假设来分析多个随机变量。\n",
    "* 期望和方差为概率分布的关键特征的概括提供了实用的度量形式。\n",
    "\n",
    "## 练习\n",
    "\n",
    "1. 我们进行了 $m=500$ 组实验，每组抽取 $n=10$ 个样本。变化 $m$ 和 $n$，观察和分析实验结果。\n",
    "2. 给定两个概率为 $P(\\mathcal{A})$ 和 $P(\\mathcal{B})$ 的事件，计算 $P(\\mathcal{A} \\cup \\mathcal{B})$ 和 $P(\\mathcal{A} \\cap \\mathcal{B})$ 的上限和下限。（提示：使用 [友元图](https://en.wikipedia.org/wiki/Venn_diagram) 来展示这些情况。）\n",
    "3. 假设我们有一系列随机变量，例如 $A$，$B$ 和 $C$，其中 $B$ 只依赖于 $A$，而 $C$ 只依赖于 $B$，你能简化联合概率 $P(A, B, C)$ 吗？（提示：这是一个 [马尔可夫链](https://en.wikipedia.org/wiki/Markov_chain)。）\n",
    "4. 在 :numref:`subsec_probability_hiv_app` 中，第一个测试更准确。为什么不运行第一个测试两次，而是同时运行第一个和第二个测试?\n"
   ]
  },
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     "pytorch"
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   },
   "source": [
    "[Discussions](https://discuss.d2l.ai/t/1762)\n"
   ]
  }
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